Basics of Probability and Statistics Week 1 Challenge

Q1.Difference in the formula of variance for Population and Sample :

• Variance - It is a way to measure spread of values in a dataset.
• Population Variance - Population variance is a type of variance that is used to determine the variability of the population data with respect to the mean.
• Sample Variance - Sample variance is used to calculate the variability in a given sample.

The formula to calculate population variance is:

σ2 = Σ (xi – μ)2 / N

where:

Σ: A symbol that means “sum”

μ: Population mean

xi: The ith element from the population

N: Population size

The formula to calculate sample variance is:

s2 = Σ (xi – x)2 / (n-1)

where:

x: Sample mean

xi: The ith element from the sample

n: Sample size

Notice that there’s only one tiny difference between the two formulas:

When we calculate population variance, we divide by N (the population size).

When we calculate sample variance, we divide by n-1 (the sample size – 1).

When calculating the sample variance, we apply something known as Bessel’s correction – which is the act of dividing by n-1.

Summary: Population variance refers to the value of variance that is calculated from population data, and sample variance is the variance calculated from sample data. Due to this value of denominator in the formula for variance in case of sample data is 'n-1', and it is 'n' for population data.

Q2. Difference between Stratified and Clustered Sampling

Factors for Comparison	Cluster Sampling	Stratified Sampling
Definition	Members of this sample are chosen from naturally divided groups called clusters, by randomly selecting elements to be a part of the sample.	Members of this sample are randomly chosen from non-overlapping, homogeneous strata.
Purpose	Cost reduction and increased efficiency	Enhanced precision and population depiction.
Sample Selection	Selection of the sample is done by randomly selected clusters and including all the members from these clusters.	Selection of the sample is done by randomly selecting members from various formed strata.
Selection of elements that form a Sample	Conjointly	Distinctively

Q3- How many different samples can be created out of population of size n ?

Ans. The size of the sample to be taken is equal to the size of the population. This means that the number of different random samples that can be taken is 1.

Q4- What is the probability of drawing 2 jacks from a pack of card ?

Ans- Probability of dwawing 2 jacks without replacement

for first jack- 4/52

for 2nd jack - 3/51

drawing two jacks = 4/52 * 3/51

= 12/2652

= 1/221

Q5- What is the probability that both the numbers on dice are same while rolling 2 dice ?

Ans- total sample = 36

outcome = 6

probability of same number on 2 dice = 6/36

= 1/6

Q6 - Why standard deviation when variance can measure the dispersion in data ?

Ans - The standard deviation is expressed in the same units as the mean is, whereas the variance is expressed in squared units.The Standard Deviation is a measure of how spread out numbers are where as the variance is the average of the squared differences from the Mean.